Minkowski world

Lorentz transformation

In physics, the Lorentz transformation converts between two different observers' measurements of space and time, where one observer is in constant motion with respect to the other. In classical physics (Galilean relativity), the only conversion believed necessary was x′ = x − vt, describing how the origin of one observer's coordinate system slides through space with respect to the other's, at speed v and along the x-axis of each frame. According to special relativity, this is only a good approximation at much smaller speeds than the speed of light, and in general the result is not just an offsetting of the x coordinates; lengths and times are distorted as well.

If space is homogeneous, then the Lorentz transformation must be a linear transformation. Also, since relativity postulates that the speed of light is the same for all observers, it must preserve the spacetime interval between any two events in Minkowski space. The Lorentz transformations describe only the transformations in which the event at x = 0, t = 0 is left fixed, so they can be considered as a rotation of Minkowski space. The more general set of transformations that also includes translations is known as the Poincaré group.

Lorentz transformation for frames in standard configuration

Assume there are two observers O and Q, each using their own Cartesian coordinate system to measure space and time intervals. O uses (t, x, y, z) and Q uses (t', x', y', z'). Assume further that the coordinate systems are oriented so that the x-axis and the x' -axis overlap, the y-axis is parallel to the y' -axis, as are the z-axis and the z' -axis. The relative velocity between the two observers is v along the common x-axis. Also assume that the origins of both coordinate systems are the same. If all these hold, then the coordinate systems are said to be in standard configuration. A
symmetric presentation
between the forward Lorentz Transformation and the inverse Lorentz Transformation
can be achieved if coordinate systems are in
symmetric configuration.
The symmetric form highlights that all physical laws should be of such a kind that
they remain unchanged under a Lorentz transformation.

The Lorentz transformation for frames in standard configuration can be shown to be:

Note that this is only the "boost", i.e. a transformation between two frames in relative motion. But the most general proper Lorentz transformation also contains a rotation of the three axes. This boost alone is given by a symmetric matrix. But the general Lorentz transformation matrix is not symmetric.

Rapidity

The Lorentz transformation can be cast into another useful form by introducing a parameter phi called the rapidity (an instance of hyperbolic angle) through the equation:

Thus, the Lorentz transformation can be seen as a hyperbolic rotation of coordinates in Minkowski space, where the rapidity phi represents the hyperbolic angle of rotation.

General boosts

For a boost in an arbitrary direction with velocity vec{v}, it is convenient to decompose the spatial vector vec{r} into components perpendicular and parallel to the velocity vec{v}: vec{r}=vec{r}_perp+vec{r}_|. Then only the component vec{r}_| in the direction of vec{v} is 'warped' by the gamma factor:

It is a result of special relativity that the interval is an invariant. That is, s^2 = s'^2 . It can be shown that this requires the coordinate transformation to be of the form

x'^mu = x^nu {Lambda^mu}_nu + C^mu .

Here, C^mu is a constant vector and {Lambda^mu}_nu a constant matrix, where we require that

eta_{munu}{Lambda^mu}_alpha{Lambda^nu}_beta = eta_{alphabeta} .

Such a transformation is called a Poincaré transformation or an inhomogeneous Lorentz transformation. The C^a represents a space-time translation. When C^a , = 0, the transformation is called an homogeneous Lorentz transformation, or simply a Lorentz transformation.

Taking the determinant of eta_{munu}{Lambda^mu}_alpha{Lambda^nu}_beta = eta_{alphabeta} gives us

det ({Lambda^a}_b) = pm 1 .

Lorentz transformations with det ({Lambda^mu}_nu)=+1 are called proper Lorentz transformations. They consist of spatial rotations and boosts and form a subgroup of the Lorentz group. Those with det({Lambda^mu}_nu)=-1 are called improper Lorentz transformations and consist of (discrete) space and time reflections combined with spatial rotations and boosts. They don't form a subgroup, as the product of any two improper Lorentz transformations will be a proper Lorentz transformation.

The composition of two Poincaré transformations is a Poincaré transformation and the set of all Poincaré transformations with the operation of composition forms a group called the Poincaré group. Under the Erlangen program, Minkowski space can be viewed as the geometry defined by the Poincaré group, which combines Lorentz transformations with translations. In a similar way, the set of all Lorentz transformations forms a group, called the Lorentz group.

A quantity invariant under Lorentz transformations is known as a Lorentz scalar.

Special relativity

One of the most astounding predictions of special relativity was the idea that time is relative. In essence, each observer's frame of reference is associated with a unique clock, the result being that time passes at different rates for different observers. This was a direct prediction from the Lorentz transformations and is called time dilation. We can also clearly see from the Lorentz transformations that the concept of simultaneity varies between reference frames. Another startling result is length contraction.

Lorentz transformations can also be used to prove that magnetic and electric fields are simply different aspects of the same force — the electromagnetic force. If we have one charge or a collection of charges which are all stationary with respect to each other, we can observe the system in a frame in which there is no motion of the charges. In this frame, there is only an electric field. If we switch to a moving frame, the Lorentz transformation will give rise to a magnetic field. These two fields are unified in the concept of the electromagnetic field.

The correspondence principle

For relative speeds much less than the speed of light, the Lorentz transformations reduce to the Galilean transformation in accordance with the correspondence principle. The correspondence limit is usually stated mathematically as v rightarrow 0, so it is usually said that non relativistic physics is a physics of "instant action at a distance" c rightarrow infty.

Many physicists, including FitzGerald, Larmor, Lorentz and Woldemar Voigt, had been discussing the physics behind these equations since 1887. Larmor and Lorentz, who believed the luminiferous aether hypothesis, were seeking the transformations under which Maxwell's equations were invariant when transformed from the ether to a moving frame. In early 1889, Heaviside had shown from Maxwell's equations that the electric field surrounding a spherical distribution of charge should cease to have spherical symmetry once the charge is in motion relative to the ether. FitzGerald then conjectured that Heaviside’s distortion result might be applied to a theory of intermolecular forces. Some months later, FitzGerald published his conjecture in Science to explain the baffling outcome of the 1887 ether-wind experiment of Michelson and Morley. This became known as the FitzGerald-Lorentz explanation of the Michelson-Morley null result, known early on through the writings of Lodge, Lorentz, Larmor, and FitzGerald. Their explanation was widely accepted as correct before 1905. Larmor gets credit for discovering the basic equations in 1897 and for being first in understanding the crucial time dilation property inherent in his equations.

Larmor's (1897) and Lorentz's (1899, 1904) final equations are algebraically equivalent to those published and interpreted as a theory of relativity by Albert Einstein (1905) but it was the French mathematician Henri Poincaré who first recognized that the Lorentz transformations have the properties of a mathematical group. Both Larmor and Lorentz discovered that the transformation preserved Maxwell's equations. Paul Langevin (1911) said of the transformation:

"It is the great merit of H. A. Lorentz to have seen that the fundamental equations of electromagnetism admit a group of transformations which enables them to have the same form when one passes from one frame of reference to another; this new transformation has the most profound implications for the transformations of space and time".

Derivation

The usual treatment (e.g., Einstein's original work) is based on the invariance of the speed of light. However, this is not necessarily the starting point: indeed (as is exposed, for example, in the second volume of the Course in Theoretical Physics by Landau and Lifshitz), what is really at stake is the locality of interactions: one supposes that the influence that one particle, say, exerts on another can not be transmitted instantaneously. Hence, there exists a theoretical maximal speed of information transmission which must be invariant, and it turns out that this speed coincides with the speed of light in vacuum. The need for locality in physical theories was already noted by Newton (see Koestler's "The Sleepwalkers"), who considered the notion of an action at a distance "philosophically absurd" and believed that gravity must be transmitted by an agent (interstellar aether) which obeys certain physical laws.

Michelson and Morley in 1887 designed an experiment, which employed an interferometer and a half-silvered mirror, that was accurate enough to detect aether flow. The mirror system reflected the light back into the interferometer. If there were an aether drift, it would produce a phase shift and a change in the interference that would be detected. However, given the results were negative, rather than validating the aether, based upon the findings aether was not confirmed. This was a major step in science that eventually resulted in Einstein's Special Theory of Relativity.

In a 1964 paper, Erik Christopher Zeeman showed that the causality preserving property, a condition that is weaker in a mathematical sense than the invariance of the speed of light, is enough to assure that the coordinate transformations are the Lorentz transformations.

From group postulates

Group Postulate Derivation

Following is a classical derivation based on group postulates and isotropy of the space.

Let us consider two inertial frames, K and K', the latter moving with velocity vec{v} with respect to the former. By rotations and shifts we can choose the z and z' axes along the relative velocity vector and also that the events (t=0,z=0) and (t'=0,z'=0) coincide. Since the velocity boost is along the z (and z') axes nothing happens to the perpendicular coordinates and we can just omit them for brevity. Now since the transformation we are looking after connects two inertial frames, it has to transform a linear motion in (t,z) into a linear motion in (t',z') coordinates. Therefore it must be a linear transformation. The general form of a linear transformation is

Let us now consider the motion of the origin of the frame K'. In the K' frame it has coordinates (t',z'=0), while in the K frame it has coordinates (t,z=vt). These two points are connected by our transformation

On the other hand the inverse transformation is the one where v is substituted by -v,

begin{bmatrix}
t z
end{bmatrix} =
begin{bmatrix}
gamma(-v) & delta(-v)
vgamma(-v) & gamma(-v)
end{bmatrix}
begin{bmatrix}
t' z'
end{bmatrix},
Now the function gamma can not depend upon the direction of v because it is apparently the factor which defines the relativistic contraction and time dilation. These two (in an isotropic world of ours) cannot depend upon the direction of v. Thus, gamma(-v)=gamma(v) and comparing the two matrices, we get

gamma^2+vdeltagamma=1. ,

At last a composition of two coordinate transformations is also a coordinate transformation, thus the product of two of our matrices should also be a matrix of the same form, in particular the diagonal elements should be equal. Calculating the product of two transformation matrices, one with v the other with v' and comparing the diagonal elements gives

frac{vgamma(v)}{delta(v)}=frac{v'gamma(v')}{delta(v')}
Since this holds for any arbitrary v and v' this combination of function must be a universal constant, one and the same for all inertial frames. Let's define this constant as frac{vgamma(v)}{delta(v)}=-c^2 where c has a dimension of velocity (we have not yet assumed, that c^2>0). Using the equation from the inverse transformation we finally get gamma=1/sqrt{1-v^2/c^2} and the transformation matrix is given by

begin{bmatrix}
t' z'
end{bmatrix} =
frac{1}{sqrt{1-v^2/c^2}}
begin{bmatrix}
1 & -v/c^2
-v & 1
end{bmatrix}
begin{bmatrix}
t z
end{bmatrix}.
Apparently c^2 cannot be negative because otherwise there would be a transformation which transforms time into spatial coordinate and vice versa. This is no good (at least in special relativity) since time can only run in the positive direction while coordinates in both. If then c^2>0 it is apparently the highest achievable velocity. Theoretically it can be either infinitely large, which gives Galilean transformation and Euclidean world with absolute time, or it can be finite, which gives Lorentz transformation and Minkowski world of special relativity. The experiment tells us that it is finite, c=299792458m/s.

From Physical Principles

Physical Derivation

The problem is usually restricted to two dimensions by using a velocity along the x axis such that the y and z coordinates do not intervene. It is similar to that of Einstein. More details may be found in
As in the Galilean transformation, the Lorentz transformation is linear : the relative velocity of the reference frames is constant. They are called inertial or Galilean reference frames. According to relativity no Galilean reference frame is privileged. Another condition is that the speed of light must be independent of the reference frame, in practice of the velocity of the light source.

Galilean reference frames

In classical kinematics, the total displacement x in the R frame is the sum of the relative displacement x' in frame R' and of the displacement x in frame R. If v is the relative velocity of R' relative to R, we have v : x = x’+vt or x’=x-vt. This relationship is linear for a constant v, that is when R and R' are Galilean frames of reference.
In Einstein's relativity, the main difference with Galilean relativity is that space is a function of time and vice-versa: t ≠ t’. The most general linear relationship is obtained with four constant coefficients, α, β, γ and v:

Speed of light independent of the velocity of the source

Light being independent of the reference frame as was shown by Michelson, we need to have x = ct if x’ = ct’. Replacing x and x' in the preceding equations, one has:

ct'=gammaleft(c-vright)t

t'=betaleft(1+alpha cright)t

Replacing t’ with the help of the second equation, the first one writes:

cbetaleft(1+alpha cright)t=gammaleft(c-vright)t

After simplification by t and dividing by cβ, one obtains:

1+alpha c=frac{gamma}{beta}(1-frac{v}{c})

Principle of relativity

According to the principle of relativity, there is no privileged Galilean frame of reference. One has to find the same Lorentz transformation from frame R to R' or from R' to R. As in the Galilean transformation, the sign of the transport velocity v has to be changed when passing from one frame to the other.

The following derivation uses only the principle of relativity which is independent of light velocity constancy.
The inverse transformation of

x'=gammaleft(x-vtright)

t'=betaleft(t+alpha xright)

is :

x=frac{1}{1-alpha v}left(frac{x'}{gamma}-frac{vt'}{beta}right)

t=frac{1}{1-alpha v}left(frac{t'}{beta}-frac{alpha x'}{gamma}right)

In accordance with the principle of relativity, the expressions of x and t are:

x=gammaleft(x'+vt'right)

t=betaleft(t'+alpha x'right)

They have to be identical to those obtained by inverting the transformation except for the sign of the velocity of transport v: